# What exactly does it mean to wrap a D-brane or a M-brane in a Riemann surface $\Sigma_g$?

+ 4 like - 0 dislike
811 views

What exactly does it mean to wrap a D-brane or an M-brane in a Riemann surface $\Sigma_g$ ($g$ is the genous)? Is there some mathematical statement? And why do we get various supersymmetric gauge theories out of this?

Is there any relation to compactifications? Is there a good place for a read on the "wrapping" of branes on Riemann surfaces?

P.S. As far as the references are concerned I would appreciate something "easier" than Nunez's and Maldacena's paper for example (http://arxiv.org/abs/hep-th/0007018). Thanks!

This post imported from StackExchange Physics at 2015-02-23 17:11 (UTC), posted by SE-user Marion

@Urs Schreiber maybe you can help out with this? A very interesting question that currently is at the center of research in susy gauge theories.

@conformal_gk, to notify a user you need to eliminate all the spaces in his/her username, in this case you need to type "at"UrsSchreiber (replace "at" by @)

+ 5 like - 0 dislike

The simplest classical picture of a brane is a manifold in space-time, representing the trajectory of a spatially extended dynamical object in string theory. What is generally called a p-brane is extended in p spatial dimensions and so its "trajectory", generally called worldvolume, is a (p+1)-dimensional manifold in spacetime. In superstring theory, the spacetime is 10 dimensional and the data of a p-brane is the data of a (p+1) dimensional submanifold of this spacetime. The simplest textbooks examples take as spacetime $\mathbb{R}^{1,9}$ and flat p-branes i.e. $\mathbb{R}^{1,p}$. But one can imagine more complicated situations. For example, to construct realistic 4-dimensional theories, one can take as spacetime $\mathbb{R}^{1,3} \times X$ where $X$ is a compact six dimensional manifold. In this case, one can consider for example p-branes of the form $\mathbb{R}^{1,3} \times Y$ where $Y$ is a submanifold of $X$. One generally says that the p-brane is wrapped around $Y$. For example, if $X$ contains a Riemann surface, one can take $Y$ to be this Riemann surface and so the p-brane is wrapped on the Riemann surface $Y$.

To summarize: a p-brane is wrapped on a manifold $Y$ if its worlvolume is a (p+1)-manifold which is a product of a flat space with $Y$.

In what preceeds, I have only considered the worldvolume manifold underlying the brane. In fact, a brane is a much richer object because it is a dynamical object in the theory. For example, for D-branes in string theory, strings can end on the D-branes and in particular we have open strings with both ends on the D-brane. The massless spectrum of these open strings contain a gauge boson and so the effective theory living on a D-brane is a gauge theory. More precisely, it is a $U(1)$ gauge theory for a unique D-brane and a $U(N)$ gauge theory if one has $N$ D-branes stacked together (in the simplest cases, say in type II superstring without spacetime singularities, without B-field...). Due to spacetime supersymmetry in string theory, it is in general a supersymmetric gauge theory. The precise number of supersymmetries preserved in the gauge theory depends precisely on the way the brane is wrapped. If the brane is wrapped on a manifold $Y$, then taking the limit when $Y$ becomes small, one obtains an effective theory in the remaining non-compact dimensions of the brane. Thus starting with a gauge theory living on a flat brane, wrapping this brane around a non-trivial manifold gives you a way to construct a new gauge theory with less spacetime dimensions and in general less supersymmetries.

In general, to preserve supersymmetries by wrapping branes on a manifold $Y$, $Y$ has to satisfy strong conditions which are mathematically very interesting (these conditions have a BPS form and are related to the theory of calibrated submanifolds. Depending on the precise context, some examples are complex submanifolds, special Lagrangian submanifolds...)

EDIT: here are some explicit examples.

1) Start with $N$ parallel coincident M5 branes. The low energy theory living on this stack of brane is a $N=(2,0)$ six dimensional gauge theory of gauge group $U(N)$. If you wrap these branes on a 2-torus, the resulting theory at low-energy in the remaining four non-compact dimensions is $N=4$ super Yang-Mills of gauge group $U(N)$. If instead of a 2-torus, one chooses a more complicated Riemann surface, one obtains a four dimensional N=2 gauge theory whose details depend on the Riemann surface. These $N=2$ 4d theories are called of class S and most of them have no lagrangian description.

2)Again $N$ $M5$ branes but in a slightly different context (we write 6=4+2 rather 6=2+4). Assume that M-theory is compactified on a Calabi-Yau 3-fold $X$. Then wrapping the M5-branes around a four dimensional submanifold of $X$ (more precisely a complex surface in $X$), gives at low energy for the remaining two non-compact dimensions a $N=(4,0)$ 2d gauge theory, which was for example used for black holes entropy calculations.

3) Consider $N$ parallel coincident D3 branes. The low energy theory living on this stack of brane is a $N=4$ super Yang-Mills of gauge group $U(N)$. Assume that type IIB string theory is compactified on a Calabi-Yau 3-fold $X$. Then, you can wrap the branes on a point in $X$, which gives something non-trivial if this point is singular in $X$, for example if $X$ is a cone over a 5-manifold $Y$ and if we put the branes at the tip of the cones. The resulting gauge theory in the non-compact four dimensions have $N=2$ or $N=1$ supersymmetries depending on the details. An AdS/CFT argument shows that these four dimensional gauge theories are dual to type IIB string theory on $AdS^5 \times Y$.

4) Consider type IIA string theory compactified on a Calabi-Yau 3-fold $X$. One can wrap $N$ D6 branes around a 3-manifold in $X$ (more precisely a special Lagrangian submanifold of $X$) and the low-energy gauge theory living in the remaining four non-compact dimensions has $N=1$ supersymmetries...

5) Consider type IIB gauge theories compactified on a complex surface $X$ resolution of an ADE singularity. One can wrap $N$ D7-branes around the non-trivial 2-cycles in $X$ and the low-energy gauge theory living in the remaining six non-compact dimensions has $N=(1,0)$ supersymmetries...

6)...

answered Feb 23, 2015 by (5,120 points)
edited Feb 24, 2015 by 40227

Would you be able to give some specific gauge theory examples?

Thanks for your really good answer. I would give a second + but it's not possible. Would you be able to give some references for the points 1 and 5 (i.e. introductory or review or lectures if possible and if they exist)? Thanks a lot.

Unfortunately, I don't know review/lectures. They probably exist but it is just that I never looked for them. The references I know are original papers. For 1), a great part of the recent activity comes from the work of Gaiotto, see for example http://arxiv.org/abs/0904.2715 For 5), they are probably many things in the F-theory litterature but what I had in mind is http://arxiv.org/abs/1312.5746

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ys$\varnothing$csOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.